To find the number of terms $n$ in an arithmetic sequence where the first term $a$ is 6, the common difference $d$ is 4, and the sum of the first $n$ terms $S_n$ is 126, we use the formula for the sum of an arithmetic sequence:

$S_n = \frac{n}{2} \left(2a + (n-1)d\right)$

Given:

• $a = 6$
• $d = 4$
• $S_n = 126$

Substitute the given values into the formula:

$126 = \frac{n}{2} \left(2 \cdot 6 + (n-1) \cdot 4\right)$

Simplify inside the parentheses:

$126 = \frac{n}{2} \left(12 + 4n - 4\right)$

$126 = \frac{n}{2} \left(8 + 4n\right)$

Multiply through by 2 to clear the fraction:

$252 = n (8 + 4n)$

Distribute $n$:

$252 = 8n + 4n^2$

Rearrange into standard quadratic form:

$4n^2 + 8n - 252 = 0$

Divide through by 4 to simplify:

$n^2 + 2n - 63 = 0$

Solve the quadratic equation using the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 2$, and $c = -63$:

$n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-63)}}{2 \cdot 1}$

$n = \frac{-2 \pm \sqrt{4 + 252}}{2}$

$n = \frac{-2 \pm \sqrt{256}}{2}$

$n = \frac{-2 \pm 16}{2}$

This gives us two solutions:

$n = \frac{-2 + 16}{2} = \frac{14}{2} = 7$

$n = \frac{-2 - 16}{2} = \frac{-18}{2} = -9$

Since $n$ must be a positive integer, we discard $n = -9$:

$n = 7$

Thus, the number of terms $n$ is 7.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. What is the significance of the common difference in an arithmetic sequence?
3. How can you determine the nth term of an arithmetic sequence?
4. Can the formula for the sum of an arithmetic sequence be used for geometric sequences?
5. What are the applications of arithmetic sequences in real-world problems?
6. How do you solve quadratic equations using the quadratic formula?
7. What other methods can be used to solve quadratic equations?
8. How does the value of the common difference affect the sum of an arithmetic sequence?

Tip: When solving quadratic equations, always check for extraneous solutions, especially in contexts where negative values may not be meaningful.